Mr. Smith has 100 students in his Algebra classes. The grades of a quiz are normally distributed with a mean of 70 and a standard deviation of 4. Sally claims that 75 students scored between 66 and 74 on this quiz. Given that this is normally distributed, is Sally correct? Justify your answer using mathematical reasoning and your knowledge of normal distributions.
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability between the two Z scores. Is it close to .75?
To determine whether Sally's claim is correct, we need to use the normal distribution properties and calculate the probability of getting a score between 66 and 74.
The first step is to convert these scores into standardized z-scores, which measure the number of standard deviations a particular score is from the mean in a normal distribution.
To find the z-score for a score of 66, we use the formula:
z = (x - μ) / σ
where x is the value we want to convert, μ is the mean, and σ is the standard deviation.
For a score of 66:
z = (66 - 70) / 4 = -1
Next, we find the z-score for a score of 74:
z = (74 - 70) / 4 = 1
Now, we need to find the probability of getting a score between these two z-scores. However, since we want the probability in a continuous range (between 66 and 74), we need to find the area under the normal curve between these two z-scores.
Using a standard normal distribution table or a statistical software, we can find the area to the left of the z-score of -1, which is approximately 0.1587.
Similarly, the area to the left of the z-score of 1 is also approximately 0.8413.
To find the area between these two z-scores, we subtract the smaller area from the larger area:
0.8413 - 0.1587 = 0.6826
This means that approximately 68.26% of students scored between 66 and 74.
Now, we compare this probability to Sally's claim that 75 students scored in this range. To do this, we need to calculate the actual number of students in this range.
Since there are 100 students in Mr. Smith's Algebra class, we can calculate the number of students within this range as a percentage of the total number of students:
0.6826 * 100 = 68.26
Therefore, based on our calculation, we would expect approximately 68 students to score between 66 and 74. This number is lower than Sally's claim of 75 students.
Given this information, we can conclude that Sally's claim is incorrect.